1 INTRODUCTION

1.1 Filtering

1.2 History of Signal Filtering

1.3 Subject Matter ofthis Book

1.4 Outline of the Book

2 FILTERING, LINEAR SYSTEMS, AND ESTIMATION

2.1 Systems, Noise, Filtering, Smoothing, and Prediction

2.2 The Gauss -Markov Discrete-time Model

2.3 Estimation Criteria

3 THE DISCRETE-TIME KALMAN FILTER

3.1 The Kalman Filter

3.2 Best Linear Estimator Property of the Kalman Filter

3.3 Identification as a Kalman Filtering Problem

3.4 Application of Kalman Filters

4 TIME-INVARIANT FILTERS

4.1 Background to Time Invariance of the Filter

4.2 Stability Properties of Linear, Discrete-time Systems

4.3 Stationary Behaviour of Linear Systems

4.4 Time Invariance and Asymptotic Stability of the Filter

4.5 Frequency Domain Formulas

5 KALMAN FILTER PROPERTIES

5.1 Introduction

5.2 Minimum Variance and Linear Minimum Variance Estimation; Orthogonality and Projection

5.3 The Innovations Sequence

5.4 The Kalman Kilter

5.5 True Filtered Estimates and the Signal-to -Noise Ratio Improvement Property

5.6 Inverse Problems: When is a Filter Optimal?

6 COMPUTATIONAL ASPECTS

6.1 Signal Model Errors, Filter Divergence, and Data Saturation

6,2 Exponential Data Weighting–A Filter with Prescribed Degree of Stability

6.3 The Matrix Inversion Lemma and the Information Filter

6.4 Sequential Processing

6.5 Square Root Filtering

6.6 The High Measurement Noise Case

6.7 Chandrasekhar – Type. Doubling. and Nonrecursive Algorithms

7 SMOOTHING OF DISCRETE-TIME SIGNALS

7.1 Introduction to Smoothing

7.2 Fixed-point Smoothing

7.3 Fixed-fag Smoothing

7.4 Fixed-interval Smoothing

8 APPLICATIONS IN NONLINEAR FILTERING

8.1 Nonlinear Filtering

8.2 The Extended Kalman Filter

8.3 A Bound Optimal Filter

8.4 Gaussian Sum Estimators

9 INNOVATIONS REPRESENTATIONS, SPECTRAL FACTORIZATION, WIENER AND LEVINSON FILTERING

9.1 Introduction

9.2 Kalman Filter Design from Covariance Data

9.3 Innovations Representations with Finite Initial Time

9.4 Stationary Innovations Representations and Spectral Factorization

9.5 Wiener Filtering

9.6 Levinson Filters

10 PARAMETER IDENTIFICATION AND ADAPTIVE ESTIMATION

10.1 Adaptive Estimation via Parallel Processing

10.2 Adaptive Estimation via Extended Least Squares

11 COLORED NOISE AND SUBOPTIMAL REDUCED ORDER FILTERS

11.1 General Approaches to Dealing with Colored Noise

11.2 Filter Design with Markov Output Noise

11.3 Filter Design with Singular or Near-singular Output Noise

11.4 Suboptimal Design Given Colored Input or Measurement Noise

11.5 Suboptimal Filter Design by Model Order Reduction

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